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I am working with Conjugate Gradient method to solve for 𝐴𝑥=𝑏, where 𝐴 is an extremely large PSD and Singular matrix. I cannot directly access the elements of 𝐴. The only thing I can do is computing 𝐴𝑣 for any vector 𝑣, which is a rather expensive operation.

Now, I would like to solve for 𝐴𝑥=0, which is effectively finding one vector in the null space of 𝐴. I understand that it is slow to solve it with CG directly. I wonder if there are other iterative methods that would give a reasonably good solution with only the access of 𝐴𝑣 very quickly?

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  • $\begingroup$ Why would CG be slow on this? The progress of CG depends on the full spectrum of A and not so much on the condition number if I remember correctly. $\endgroup$
    – Dirk
    Commented Mar 2, 2023 at 21:10
  • $\begingroup$ I mean CG is for sure the fastest iterative method to solve for any general equations Ax=b (A being PSD). However, for the particular case Ax=0, CG is obviously not utilising the information of b = 0. I wonder if I could incorporate this information to improve the convergence speed of CG, or, if there are other methods that are particularly efficient in solving Ax = 0. $\endgroup$
    – HANDSOMEJACKANDY
    Commented Mar 2, 2023 at 21:16

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